\(\int (a+\frac {b}{x})^m (c+d x)^2 \, dx\) [597]

Optimal result

Integrand size = 17, antiderivative size = 134 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\frac {c^2 \left (a+\frac {b}{x}\right )^{1+m} x^2}{b (1-m)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+m} x^3}{3 a}-\frac {b \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (2-3 m+m^2\right )\right ) \left (a+\frac {b}{x}\right )^{1+m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,1+\frac {b}{a x}\right )}{3 a^4 \left (1-m^2\right )} \] Output:

c^2*(a+b/x)^(1+m)*x^2/b/(1-m)+1/3*d^2*(a+b/x)^(1+m)*x^3/a-1/3*b*(6*a^2*c^2 
-6*a*b*c*d*(1-m)+b^2*d^2*(m^2-3*m+2))*(a+b/x)^(1+m)*hypergeom([3, 1+m],[2+ 
m],1+b/a/x)/a^4/(-m^2+1)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\frac {\left (a+\frac {b}{x}\right )^m (b+a x) \left (a^2 d (1+m) x^2 (b d (-2+m)+2 a (3 c+d x))-b \left (6 a^2 c^2+6 a b c d (-1+m)+b^2 d^2 \left (2-3 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,1+\frac {b}{a x}\right )\right )}{6 a^4 (1+m) x} \] Input:

Integrate[(a + b/x)^m*(c + d*x)^2,x]
 

Output:

((a + b/x)^m*(b + a*x)*(a^2*d*(1 + m)*x^2*(b*d*(-2 + m) + 2*a*(3*c + d*x)) 
 - b*(6*a^2*c^2 + 6*a*b*c*d*(-1 + m) + b^2*d^2*(2 - 3*m + m^2))*Hypergeome 
tric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)]))/(6*a^4*(1 + m)*x)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {941, 948, 100, 87, 75}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^2 \left (a+\frac {b}{x}\right )^m \, dx\)

\(\Big \downarrow \) 941

\(\displaystyle \int x^2 \left (\frac {c}{x}+d\right )^2 \left (a+\frac {b}{x}\right )^mdx\)

\(\Big \downarrow \) 948

\(\displaystyle -\int \left (a+\frac {b}{x}\right )^m \left (\frac {c}{x}+d\right )^2 x^4d\frac {1}{x}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {d^2 x^3 \left (a+\frac {b}{x}\right )^{m+1}}{3 a}-\frac {\int \left (a+\frac {b}{x}\right )^m \left (\frac {3 a c^2}{x}+d (6 a c-b d (2-m))\right ) x^3d\frac {1}{x}}{3 a}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {d^2 x^3 \left (a+\frac {b}{x}\right )^{m+1}}{3 a}-\frac {\frac {\left (6 a^2 c^2-b d (1-m) (6 a c-b d (2-m))\right ) \int \left (a+\frac {b}{x}\right )^m x^2d\frac {1}{x}}{2 a}-\frac {d x^2 \left (a+\frac {b}{x}\right )^{m+1} (6 a c-b d (2-m))}{2 a}}{3 a}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {d^2 x^3 \left (a+\frac {b}{x}\right )^{m+1}}{3 a}-\frac {\frac {b \left (a+\frac {b}{x}\right )^{m+1} \left (6 a^2 c^2-b d (1-m) (6 a c-b d (2-m))\right ) \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,\frac {b}{a x}+1\right )}{2 a^3 (m+1)}-\frac {d x^2 \left (a+\frac {b}{x}\right )^{m+1} (6 a c-b d (2-m))}{2 a}}{3 a}\)

Input:

Int[(a + b/x)^m*(c + d*x)^2,x]
 

Output:

(d^2*(a + b/x)^(1 + m)*x^3)/(3*a) - (-1/2*(d*(6*a*c - b*d*(2 - m))*(a + b/ 
x)^(1 + m)*x^2)/a + (b*(6*a^2*c^2 - b*d*(6*a*c - b*d*(2 - m))*(1 - m))*(a 
+ b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)])/(2*a^3*(1 
+ m)))/(3*a)
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 941

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym 
bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, 
 n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !IntegerQ[p])
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \left (a +\frac {b}{x}\right )^{m} \left (d x +c \right )^{2}d x\]

Input:

int((a+b/x)^m*(d*x+c)^2,x)
 

Output:

int((a+b/x)^m*(d*x+c)^2,x)
 

Fricas [F]

\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\int { {\left (d x + c\right )}^{2} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:

integrate((a+b/x)^m*(d*x+c)^2,x, algorithm="fricas")
 

Output:

integral((d^2*x^2 + 2*c*d*x + c^2)*((a*x + b)/x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.93 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87 \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\frac {b^{m} c^{2} x^{1 - m} \Gamma \left (1 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 1 - m \\ 2 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac {2 b^{m} c d x^{2 - m} \Gamma \left (2 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 2 - m \\ 3 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} + \frac {b^{m} d^{2} x^{3 - m} \Gamma \left (3 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 3 - m \\ 4 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (4 - m\right )} \] Input:

integrate((a+b/x)**m*(d*x+c)**2,x)
 

Output:

b**m*c**2*x**(1 - m)*gamma(1 - m)*hyper((-m, 1 - m), (2 - m,), a*x*exp_pol 
ar(I*pi)/b)/gamma(2 - m) + 2*b**m*c*d*x**(2 - m)*gamma(2 - m)*hyper((-m, 2 
 - m), (3 - m,), a*x*exp_polar(I*pi)/b)/gamma(3 - m) + b**m*d**2*x**(3 - m 
)*gamma(3 - m)*hyper((-m, 3 - m), (4 - m,), a*x*exp_polar(I*pi)/b)/gamma(4 
 - m)
 

Maxima [F]

\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\int { {\left (d x + c\right )}^{2} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:

integrate((a+b/x)^m*(d*x+c)^2,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^2*(a + b/x)^m, x)
 

Giac [F]

\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\int { {\left (d x + c\right )}^{2} {\left (a + \frac {b}{x}\right )}^{m} \,d x } \] Input:

integrate((a+b/x)^m*(d*x+c)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((d*x + c)^2*(a + b/x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\int {\left (a+\frac {b}{x}\right )}^m\,{\left (c+d\,x\right )}^2 \,d x \] Input:

int((a + b/x)^m*(c + d*x)^2,x)
 

Output:

int((a + b/x)^m*(c + d*x)^2, x)
 

Reduce [F]

\[ \int \left (a+\frac {b}{x}\right )^m (c+d x)^2 \, dx=\frac {6 \left (a x +b \right )^{m} a^{2} c^{2} x +6 \left (a x +b \right )^{m} a^{2} c d \,x^{2}+2 \left (a x +b \right )^{m} a^{2} d^{2} x^{3}+6 \left (a x +b \right )^{m} a b c d m x +\left (a x +b \right )^{m} a b \,d^{2} m \,x^{2}+\left (a x +b \right )^{m} b^{2} d^{2} m^{2} x -2 \left (a x +b \right )^{m} b^{2} d^{2} m x +6 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a^{2} b \,c^{2} m +6 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a \,b^{2} c d \,m^{2}-6 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) a \,b^{2} c d m +x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{3} d^{2} m^{3}-3 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{3} d^{2} m^{2}+2 x^{m} \left (\int \frac {\left (a x +b \right )^{m}}{x^{m} a x +x^{m} b}d x \right ) b^{3} d^{2} m}{6 x^{m} a^{2}} \] Input:

int((a+b/x)^m*(d*x+c)^2,x)
 

Output:

(6*(a*x + b)**m*a**2*c**2*x + 6*(a*x + b)**m*a**2*c*d*x**2 + 2*(a*x + b)** 
m*a**2*d**2*x**3 + 6*(a*x + b)**m*a*b*c*d*m*x + (a*x + b)**m*a*b*d**2*m*x* 
*2 + (a*x + b)**m*b**2*d**2*m**2*x - 2*(a*x + b)**m*b**2*d**2*m*x + 6*x**m 
*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*a**2*b*c**2*m + 6*x**m*int((a*x + 
 b)**m/(x**m*a*x + x**m*b),x)*a*b**2*c*d*m**2 - 6*x**m*int((a*x + b)**m/(x 
**m*a*x + x**m*b),x)*a*b**2*c*d*m + x**m*int((a*x + b)**m/(x**m*a*x + x**m 
*b),x)*b**3*d**2*m**3 - 3*x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*b** 
3*d**2*m**2 + 2*x**m*int((a*x + b)**m/(x**m*a*x + x**m*b),x)*b**3*d**2*m)/ 
(6*x**m*a**2)